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Further enumeration results concerning a recent equivalence of restricted inversion sequences

Mansour, Toufik ; Shattuck, Mark

Discrete Mathematics and Theoretical Computer Science, 2022-01, Vol.24 (1), p.1-28 [Periódico revisado por pares]

Nancy: DMTCS

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Título:
Further enumeration results concerning a recent equivalence of restricted inversion sequences
Autor: Mansour, Toufik ; Shattuck, Mark
Assuntos: 05a05 ; 2021. 2010 mathematics subject classification. 05a15 ; 65q20 ; 65q30 pattern avoidance ; [math.math-co]mathematics [math]/combinatorics [math.co] ; [math.math-mp]mathematics [math]/mathematical physics [math-ph] ; Algorithms ; combinatorial statistic ; Combinatorics ; Enumeration ; Equivalence ; Functional equations ; inversion sequence ; Inversions (Geometry) ; july 22 ; kernel method ; Mathematical Physics ; Mathematical research ; Mathematics ; Sequences ; Sequences (Mathematics)
É parte de: Discrete Mathematics and Theoretical Computer Science, 2022-01, Vol.24 (1), p.1-28
Descrição: Let asc and desc denote respectively the statistics recording the number of ascents or descents in a sequence having non-negative integer entries. In a recent paper by Andrews and Chern, it was shown that the distribution of asc on the inversion sequence avoidance class $I_n(\geq,\neq,>)$ is the same as that of $n-1-\text{asc}$ on the class $I_n(>,\neq,\geq)$, which confirmed an earlier conjecture of Lin. In this paper, we consider some further enumerative aspects related to this equivalence and, as a consequence, provide an alternative proof of the conjecture. In particular, we find recurrence relations for the joint distribution on $I_n(\geq,\neq,>)$ of asc and desc along with two other parameters, and do the same for $n-1-\text{asc}$ and desc on $I_n(>,\neq,\geq)$. By employing a functional equation approach together with the kernel method, we are able to compute explicitly the generating function for both of the aforementioned joint distributions, which extends (and provides a new proof of) the recent result $|I_n(\geq,\neq,>)|=|I_n(>,\neq,\geq)|$. In both cases, an algorithm is formulated for computing the generating function of the asc distribution on members of each respective class having a fixed number of descents.
Editor: Nancy: DMTCS
Idioma: Inglês

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Further enumeration results concerning a recent equivalence of restricted inversion sequences

Mansour, Toufik ; Shattuck, Mark

Nancy: DMTCS

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